We construct the Lagrangian for an effective theory of highly energetic quarks with energy Q, interacting with collinear and soft gluons. This theory has two low energy scales, the transverse momentum of the collinear particles, p ⊥ , and the scale p 2 ⊥ /Q. The heavy to light currents are matched onto operators in the effective theory at one-loop and the renormalization group equations for the corresponding Wilson coefficients are solved. This running is used to sum Sudakov logarithms in inclusive B → X s γ and B → X u ℓν decays. We also show that the interactions with collinear gluons preserve the relations for the soft part of the form factors for heavy to light decays found by Charles et al., establishing these relations in the large energy limit of QCD.

The factorization of soft and ultrasoft gluons from collinear particles is shown at the level of operators in an effective field theory. Exclusive hadronic factorization and inclusive partonic factorization follow as special cases. The leading order Lagrangian is derived using power counting and gauge invariance in the effective theory. Several species of gluons are required, and softer gluons appear as background fields to gluons with harder momenta. Two examples are given: the factorization of soft gluons in B → Dπ, and the soft-collinear convolution for the B → X s γ spectrum.

We consider processes which produce final state hadrons whose energy is much greater than their mass. In this limit interactions involving collinear fermions and gluons are constrained by a symmetry, and we give a general set of rules for constructing leading and subleading invariant operators. Wilson coefficients C(µ,P) are functions of a label operatorP, and do not commute with collinear fields. The symmetry is used to reproduce a two-loop result for factorization in B → Dπ in a simple way.

In this paper we show how gauge symmetries in an effective theory can be used to simplify proofs of factorization formulas in highly energetic hadronic processes. We use the soft-collinear effective theory, generalized to deal with back-to-back jets of collinear particles. Our proofs do not depend on the choice of a particular gauge, and the formalism is applicable to both exclusive and inclusive factorization. As examples we treat the -␥ form factor (␥␥*→ 0 ), light meson form factors (␥*M →M ), as well as deep inelastic scattering (e Ϫ p→e Ϫ X), the Drell-Yan process (pp →Xl ϩ l Ϫ ), and deeply virtual Compton scattering

We construct jet observables for energetic top quarks that can be used to determine a short-distance top quark mass from reconstruction in e e ÿ collisions with accuracy better than QCD . Using a sequence of effective field theories we connect the production energy, mass, and top width scales, Q m ÿ, for the top jet cross section, and derive a QCD factorization theorem for the top invariant mass spectrum. Our analysis accounts for s corrections from the production and mass scales, corrections due to constraints in defining invariant masses, nonperturbative corrections from the cross talk between the jets, and s corrections to the Breit-Wigner line shape. This paper mainly focuses on deriving the factorization theorem for hemisphere invariant mass distributions and other event shapes in e e ÿ collisions applicable at a future linear collider. We show that the invariant mass distribution is not a simple Breit-Wigner function involving the top width. Even at leading order it is shifted and broadened by nonperturbative soft QCD effects. We predict that the invariant mass peak position increases linearly with Q=m due to these nonperturbative effects. They are encoded in terms of a universal soft function that also describes soft effects for massless dijet events. In a future paper we compute s corrections to the jet invariant mass spectrum, including a summation of large logarithms between the scales Q, m, and ÿ.

We give a factorization formula for the e + e − thrust distribution dσ/dτ with τ = 1 − T based on soft-collinear effective theory. The result is applicable for all τ , i.e. in the peak, tail, and fartail regions. The formula includes O(α 3 s ) fixed-order QCD results, resummation of singular partonic α j s ln k (τ )/τ terms with N 3 LL accuracy, hadronization effects from fitting a universal nonperturbative soft function defined in field theory, bottom quark mass effects, QED corrections, and the dominant top mass dependent terms from the axial anomaly. We do not rely on Monte Carlo generators to determine nonperturbative effects since they are not compatible with higher order perturbative analyses. Instead our treatment is based on fitting nonperturbative matrix elements in field theory, which are moments Ωi of a nonperturbative soft function. We present a global analysis of all available thrust data measured at center-of-mass energies Q = 35 to 207 GeV in the tail region, where a two parameter fit to αs(mZ) and the first moment Ω1 suffices. We use a short distance scheme to define Ω1, called the R-gap scheme, thus ensuring that the perturbative dσ/dτ does not suffer from an O(ΛQCD) renormalon ambiguity. We find αs(mZ) = 0.1135 ± (0.0002)expt ± (0.0005) hadr ± (0.0009)pert, with χ 2 /dof = 0.91, where the displayed 1-sigma errors are the total experimental error, the hadronization uncertainty, and the perturbative theory uncertainty, respectively. The hadronization uncertainty in αs is significantly decreased compared to earlier analyses by our two parameter fit, which determines Ω1 = 0.323 GeV with 16% uncertainty.

We study proton-(anti)proton collisions at the LHC or Tevatron in the presence of experimental restrictions on the hadronic final state and for generic parton momentum fractions. At the scale Q of the hard interaction, factorization does not yield standard parton distribution functions (PDFs) for the initial state. The measurement restricting the hadronic final state introduces a new scale µB ≪ Q and probes the proton prior to the hard collision. This corresponds to evaluating the PDFs at the scale µB. After the proton is probed, the incoming hard parton is contained in an initialstate jet, and the hard collision occurs between partons inside these jets rather than inside protons. The proper description of such initial-state jets requires "beam functions". At the scale µB, the beam function factorizes into a convolution of calculable Wilson coefficients and PDFs. Below µB, the initial-state evolution is described by the usual PDF evolution which changes x, while above µB it is governed by a different renormalization group evolution that sums double logarithms of µB/Q and leaves x fixed. As an example, we prove a factorization theorem for "isolated Drell-Yan", pp → Xℓ + ℓ − where X is restricted to have no central jets. We comment on the extension to cases where the hadronic final state contains a certain number of isolated central jets.

We study a Lagrangian formalism that avoids double counting in effective field theories where distinct fields are used to describe different infrared momentum regions for the same particle. The formalism leads to extra subtractions in certain diagrams and to a new way of thinking about factorization of modes in quantum field theory. In non-relativistic field theories, the subtractions remove unphysical pinch singularities in box type diagrams, and give a derivation of the known pull-up mechanism between soft and ultrasoft fields which is required by the renormalization group evolution. In a field theory for energetic particles, the soft-collinear effective theory (SCET), the subtractions allow the theory to be defined with different infrared and ultraviolet regulators, remove double counting between soft, ultrasoft, and collinear modes, and give results which reproduce the infrared divergences of the full theory. Our analysis shows that convolution divergences in factorization formulae occur due to an overlap of momentum regions. We propose a method that avoids this double counting, which helps to resolve a long standing puzzle with singularities in collinear factorization in QCD. The analysis gives evidence for a factorization in rapidity space in exclusive decays.